# Factors, Multiples and Primes - Short Problems

### Four or Five

##### Stage: 3 Short Challenge Level:

Weekly Problem 19 - 2012
The diagram shows a large rectangle composed of 9 smaller rectangles. If each of these rectangles has integer sides, what could the area of the large rectangle be?

### Seven from Nine

##### Stage: 3 Short Challenge Level:

Weekly Problem 21 - 2013
In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?

### Factor Trio

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 45 - 2013
Which of the numbers shown is the product of exactly 3 distinct prime factors?

### Back of the Queue

##### Stage: 3 Short Challenge Level:

Weekly Problem 48 - 2013
What is the remainder when the number 743589×301647 is divided by 5?

### Calculation 2000

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?

### Weekly Problem 10 - 2008

##### Stage: 3 and 4 Challenge Level:

If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?

### Weekly Problem 14 - 2008

##### Stage: 3 and 4 Challenge Level:

The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?

### Weekly Problem 22 - 2008

##### Stage: 3 and 4 Challenge Level:

The following sequence continues indefinitely... Which of these integers is a multiple of 81?

### Weekly Problem 26 - 2008

##### Stage: 3 and 4 Challenge Level:

If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

### Weekly Problem 41 - 2008

##### Stage: 3 Challenge Level:

How many pairs of numbers of the form x, 2x+1 are there in which both numbers are prime numbers less than 100?

### School Dinners

##### Stage: 3 Short Challenge Level:

Weekly Problem 1 - 2009
Our school dinners offer the same choice each day, and each day I try a new option. How long will it be before I eat the same meal again?

### Little Goldbach

##### Stage: 3 Short Challenge Level:

Weekly Problem 11 - 2009
How many of the numbers 1 to 20 are not the sum of two primes?

### Trailing Zeros

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 13 - 2009
How many zeros does 50! have at the end?

### Weekly Problem 41 - 2009

##### Stage: 2 and 3 Short Challenge Level:

At a cinema a child's ticket costs £$4.20$ and an adult's ticket costs £$7.70$. How much did is cost this group of adults and children to see a film?

### Weekly Problem 45 - 2009

##### Stage: 2 and 3 Short Challenge Level:

Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

### Three Primes

##### Stage: 3 Short Challenge Level:

Weekly Problem 6 - 2010
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?

### Square LCM

##### Stage: 3 Short Challenge Level:

Weekly Problem 7 - 2010
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

### Factorised Factorial

##### Stage: 3 Short Challenge Level:

Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

### 17s and 23s

##### Stage: 3 Short Challenge Level:

Weekly Problem 22 - 2010
Can you form this 2010-digit number...

### HCF Expression

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 30 - 2010
Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

### Product 100

##### Stage: 3 Short Challenge Level:

Weekly Problem 38 - 2010
The product of four different positive integers is 100. What is the sum of these four integers?

### Factor List

##### Stage: 3 Short Challenge Level:

Weekly Problem 48 - 2010
Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

### Adjacent Factors

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 5 - 2011
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

### Sticky Fingers

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 7 - 2011
Roo wants to puts stickers on the cuboid he has made from little cubes. Will he have any stickers left over?

### Square Product

##### Stage: 4 Short Challenge Level:

Weekly Problem 10 - 2011
Will this product give a perfect square?

### Indivisible

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 25 - 2011
Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?

### Many Clues, One Answer

##### Stage: 3 Short Challenge Level:

Weekly Problem 35 - 2011
You are given lots of clues about a number. Can you work out what it is?

### Divisible Palindrome

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 5 - 2014
What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?

### Digital Division

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 15 - 2014
How many three digit numbers formed with three different digits from 0, 1, 2, 3 and 5 are divisible by 6?

### Almost a Million

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 26 - 2014
Which of the given numbers are divisible by 6?

### Supercomputer

##### Stage: 4 Short Challenge Level:

Weekly Problem 28 - 2014
What is the units digit of the given expression?

### One Less Remainder

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 39 - 2014
A whole number less than 100 gives remainders of 2, 3 and 4 when divided by 3, 4 and 5. What is the remainder when it is divided by 7?

### Primes and Six

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 1 - 2015
If $p$ and $q$ are prime numbers greater than $3$ and $q=p+2$, prove that $pq+1$ is divisible by $36$.

### Factor Sum

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 4 - 2015
Given any positive integer n, Paul adds together the factors of n, apart from n itself. Which of the numbers 1, 3, 5, 7 and 9 can never be Paul's answer?

### Tricky Customer

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 6 - 2015
Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?

### Added Power

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 33 - 2015
How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?

### Last-but-one

##### Stage: 3 Short Challenge Level:

Weekly Problem 1 - 2016
What is the last-but-one digit of 99! ?

### Birthday Tables

##### Stage: 3 Short Challenge Level:

Weekly Problem 2 - 2016
How many tables of each type does Mark need at his birthday party?

### Simon's Age

##### Short Challenge Level:

Weekly Problem 3 - 2016
How many times has Simon's age changed from a square to a prime?

### Grandma's Cake

##### Stage: 3 Short Challenge Level:

Weekly Problem 7 - 2016
What is the smallest number of pieces grandma should cut her cake into to guarentee each grandchild gets the same amount of cake and none is left over.

### Divisible by Its Digit Sum

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 18 - 2016
The year 2010 is one in which the sum of the digits is a factor of the year itself. What is the next year that has the same property?

### End of a Prime

##### Stage: 3 Short Challenge Level:

Weekly Problem 25 - 2016
A list is made of every digit that is the units digit of at least one prime number. How many digits appear in the list?

### Stamp Collecting

##### Stage: 3 Short Challenge Level:

Weekly Problem 28 - 2016
Last week, Tom and Sophie bought some stamps for their collections. Each stamp Tom bought cost him £1.10, whilst Sophie paid 70p for each of her stamps. Between them they spent exactly £10. How many stamps did they buy in total?

### Threes and Fours

##### Stage: 3 Short Challenge Level:

Weekly Problem 32 - 2016
What is the smallest integer which has every digit a 3 or a 4 and is divisible by both 3 and 4?

### Square Percentage

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 36 - 2016
What percentage of the integers between 1 and 10,000 are square numbers?

### 354972

##### Stage: 3 Short Challenge Level:

Weekly Problem 42 - 2016
What is the remainder when 354972 is divided by 7?

### Knockdown

##### Stage: 3 Short Challenge Level:

Weekly Problem 51 - 2016
Pegs numbered 1 to 50 are placed in a row. Alternate pegs are knocked down, and this process is repeated. What is the number of the last peg to be knocked down?

### Multiplication Table Puzzle

##### Stage: 3 Short Challenge Level:

Weekly Problem 6 - 2017
In the multiplication table on the right, only some of the numbers have been given. What is the value of A+B+C+D+E?

### Suit Sequence

##### Stage: 3 Short Challenge Level:

Weekly Problem 8 - 2017
A pattern repeats every six symbols. What are the 100th and 101st symbols?

### Peter's Primes

##### Stage: 3 Short Challenge Level:

Weekly Problem 22 - 2017
Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?

### Triangular Algebra

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 26 - 2017
The angles in the triangle are shown in the diagram in terms of x and y. If x and y are positive integers, what is the value of x+y?

### Primes 6 Apart

##### Stage: 3 Short Challenge Level:

Weekly Problem 30 - 2017
The prime numbers p and q are the smallest primes that differ by 6. What is the value of p+q?

### Smallest Abundant Number

##### Stage: 3 Short Challenge Level:

Weekly Problem 34 - 2017
An abundant number is a positive integer N such that the sum of the factors of N is larger than 2N. What is the smallest abundant number?

### Ones, Twos and Threes

##### Stage: 3 Short Challenge Level:

Weekly Problem 5 - 2017
Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?

### Square Total

##### Stage: 3 Short Challenge Level:

Weekly Problem 13 - 2017
Anastasia thinks of a number. Each of her friends performs an operation on it. The total of these is a square number. What is the smallest number Anastasia could have thought of?

### What's on the Back?

##### Stage: 3 Short Challenge Level:

Weekly Problem 21 - 2017
Four cards have a number on one side and a phrase on the back. On each card, the number does not have the property described on the back. What do the four cards have on them?

### Common Remainder

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 35 - 2017
144 divided by n leaves a remainder of 11. 220 divided by n also leaves a remainder of 11. What is n?

### Long List

##### Stage: 3 Short Challenge Level:

Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?